LAP Lambert Academic Publishing Problems about the Axiom of Choice: In Defense of Platonic Realism in Mathematics

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. The controversy over the Axiom of Choice (AC) is a case in point. Due to its non- constructive nature, the AC has seemingly unpleasant consequences. It leads to the existence of non- Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the latter is so called in the sense that it is a counter-intuitive theorem. To see that mathematical truths are of non- constructive nature, I draw upon Gödel's Incompleteness Theorems. The Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam's arguments assume a clear meaning. According to them, the AC depends for its truth-value upon the model in which it is placed. In my view, however, this shows a limitation of formal methods. In response to Benacerraf's challenge to Platonism, the book concludes that in mathematics, as distinct from natural sciences, Platonists see a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. About the Author Ph.D. in Philosophy from Florida State University, U.S. B.A. and M.A. in the same subject both from Kyoto University, Japan. His research focus is in the areas of logic, the philosophy of mathematics, and the philosophy of science in general.